Next we use the result of Proposition \(\ref {fix-subcode order 2}\)  to find a lower bound for \(d(C_o)\) . Note that \({\rm ord}_n(4)=26\)  and \(4^{13}\equiv -1 \pmod {157}\) , and we use the inequality \(d( (C_{o})_{-1})/2+1\le d(C_o)\) , where \((C_{o})_{-1}\)  is the fixed subcode of \(C_o\)  by the action of multiplier \(\mu _{-1}\) .
Our computation done in Magma [1]} shows that \(d((C_{o})_{-1})=36\) . Hence \(d(C_o)\ge 19\)  and the fact that \(d\)  is even shows that \(Q\)  has parameters \([[158,0,d\ge 20]]\) . Thus \(Q\)  is a new quantum code with a better minimum distance in comparison with the previous best known code with the same length and dimension which had minimum distance 19.
